In this talk, I will give a historical overview of the discovery of the first binary pulsar and the indirect observation of gravitational waves in the late 1970's. I then present the latest results on binary pulsar timing, focusing on the high-precision tests of different gravity theories, and a deep study of the nature of gravitational waves. Finally, I compare our results with what we might be able to gleam from future ground-based detectors of gravitational waves, like Advanced LIGO and the Einstein Telescope.